Let \(R=\lbrace (x,\mu )\in X\times \operatorname{\mathcal {M}}(X):\mu \in V(x)\rbrace \subseteq X\times \operatorname{\mathcal {M}_{\mathit {T}} }(X)\) . We have
\(R=\bigcap _{m=1}^\infty \bigcap _{N=1}^\infty \bigcup _{n=N}^\infty \lbrace (x,\mu ) \in X\times \operatorname{\mathcal {M}_{\mathit {T}} }(X) :D(\mu ,\operatorname{\mathcal {E}}(x,n))<1/m\rbrace .\) 
Clearly, the set \(\lbrace (x,\mu ) \in X\times \operatorname{\mathcal {M}_{\mathit {T}} }(X) :D(\mu ,\operatorname{\mathcal {E}}(x,n))<1/m\rbrace \)  is open in \(X\times \operatorname{\mathcal {M}_{\mathit {T}} }(X)\)  for every \(n,m>0\) , so \(\lbrace (x,\mu )\in X\times \operatorname{\mathcal {M}}(X):\mu \in V(x)\rbrace \subseteq X\times \operatorname{\mathcal {M}_{\mathit {T}} }(X)\)  is Borel.
Since \({U}{}G\)  is a projection of the set \(R\cap (X\times U)\)  onto the first coordinate, we get that \({U}{}G\)  is \(\Sigma ^1_1\)  (see [1]}).
